Nonviscously Damped Research Articles

A single-step recursive algorithm of the convolution integral for computing non-viscous damping forces in dynamic analyses

The non-viscous damping force features the convolution integral of the velocity and the kernel function, causing the difficulty of the solutions of the system equation of motion due to the nonlocal time dependence. In accordance with the linear time-invariant system theory in signal processing, this paper presents a single-step recursive algorithm of convolution integral for efficiently computing non-viscous damping forces. The algorithm parameters are effectively identified from the discrete responses of the kernel function to the rectangular unit impulse through digital filter technologies. The proposed algorithm is compatible with the solution schemes based on existing time integration methods, which handle the nonlocal dependence using internal recursive support vectors while leaving the order of the equation of motion unchanged. The central difference method, the Newmark-ß method, and the Bathe method are employed to illustrate the use of the algorithm, proving to retain the stability characteristics of time integration methods. The single and multiple DOF examples show that the simulated results based on the compatible solution schemes compare well with those built on other existing methods, verifying high accuracy and efficiency. The proposed algorithm thus extends the application range of existing time integration methods from the viscously damped system to the non-viscously damped one, fully demonstrating its generalizability.

Wave dispersion and dissipation performance of locally resonant acoustic metamaterials using an internal variable model

Engineering structures for different dispersion and dissipation levels of wave propagation use internal variable models, which may enhance the performance of acoustic metamaterials (AMMs). In this study, the wave dispersion and dissipation performance of AMMs is studied using an anelastic displacement fields (ADF) model. A symmetric state-space method based on Floquet-Bloch’s theorem for a nonviscously damped unit cell is developed. The study also constructs Bloch’s eigenvalue problems built from the symmetric state-space formulation to obtain the wavevector-dependent damped frequency and damping ratio for wave propagation analysis of periodic structures. The effects of wave dispersion and dissipation on the performance of AMMs are studied by using two numerical examples of mass-in-mass lattice systems containing multiple resonators. It is shown that nonviscous damping increases the wave dispersion performance of AMM. It is also shown that the metadamping phenomenon enhances the wave dissipation performance of AMM. It is demonstrated that the new method in symmetric form is applicable for performance analysis of periodic phononic crystal.

An iterative method for exact eigenvalues and eigenvectors of general nonviscously damped structural systems

A novel and computationally efficient iterative method is proposed for the exact eigenvalues and eigenvectors of nonviscously damped vibration systems. General nonviscous damping model is assumed in which damping forces depend on the past motion history via convolution integrals over exponentially decaying kernel functions. The presence of nonviscous damping leads to complex frequency dependent eigenvalue problem whose solution requires either extensively augmented state-space formulation or iterative full complex eigensolutions on mode by mode basis, both of which are computationally expensive when practical systems with large dimensions are considered. By simply solving the eigenvalue problem of the underlying undamped vibration system, the real eigenvalues and eigenvectors can then be combined with the nonviscous damping matrix to develop an iterative procedure from which required complex eigenvalues and eigenvectors of the damped system can be computed. First order perturbation is employed to further improve the starting estimates of the desired eigenvalues and eigenvectors and the convergence is generally very fast. The proposed method is mathematically developed based on progressive eigensensitivity analysis and convergence is ensured when the norm of the damping matrix is of second order when compared with that of the stiffness matrix. Representative numerical examples of discrete mass spring damping model as well as practical finite element model with nonviscous damping are given to demonstrate and validate the accuracy and efficiency of the proposed iterative method.

Stochastic Stability Analysis of Coupled Viscoelastic Systems with Nonviscously Damping Driven by White Noise

Nonviscously damped structural system has been raised in many engineering fields, in which the damping forces depend on the past time history of velocities via convolution integrals over some kernel functions. This paper investigates stochastic stability of coupled viscoelastic system with nonviscously damping driven by white noise through moment Lyapunov exponents and Lyapunov exponents. Using the coordinate transformation, the coupled Itô stochastic differential equations of the norm of the response and angles process are obtained. Then the problem of the moment Lyapunov exponent is transformed to the eigenvalue problem, and then the second-perturbation method is used to derive the moment Lyapunov exponent of coupled stochastic system. Lyapunov exponent also can be obtained according to the relationship between moment Lyapunov exponent and Lyapunov exponent. Finally, the effects of various physical quantities of stochastic coupled system on the stochastic stability are discussed in detail. These results are validated by the direct Monte Carlo simulation technique.

Critical damping in nonviscously damped linear systems

• Viscoelastic structures are characterized by a frequency dependent damping matrix. • Critically damped modes of nonviscous systems are characterized. • Mathematical conditions to find critical surfaces in nonviscous systems are presented. • A new numerical method to compute critical curves in nonviscous systems is developed. • Critical surfaces of single and multiple degrees of freedom systems are presented. In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-history of the response velocity, mathematically represented by convolution integrals involving hereditary functions. Combination of damping parameters in the dissipative model can lead the system to be overdamped in some (or all) modes. In the domain of the damping parameters, the thresholds between induced oscillatory and non-oscillatory motion are named critical damping surfaces (or critical manifolds, since several parameters can be involved). In this paper the theoretical foundations to determine critical damping surfaces in nonviscously damped systems are established. In addition, a numerical method to obtain critical curves is developed. The approach is based on the transformation of the algebraic equations , which define implicitly the critical curves, into a system of differential equations. The derivations are validated with three numerical methods covering single and multiple degree of freedom systems.

Proposal of a Viscous Model for Nonviscously Damped Beams Based on Fractional Derivatives

Viscoelastic materials are widely used in structural dynamics for the control of the vibrations and energy dissipation. They are characterized by damping forces that depend on the history of the velocity response via hereditary functions involved in convolution integrals, leading to a frequency‐dependent damping matrix. In this paper, one‐dimensional beam structures with viscoelastic materials based on fractional derivatives are considered. In this work, the construction of a new equivalent viscous system with fictitious parameters but capable of reproducing the response of the viscoelastic original one with acceptable accuracy is proposed. This allows us to take advantage of the well‐known available numerical tools for viscous systems and use them to find response of viscoelastic structures. The process requires the numerical computation of complex frequencies. The new fictitious viscous parameters are found to be matching the information provided by the frequency response functions. New mass, damping, and stiffness matrices are found, which in addition have the property of proportionality, so they become diagonal in the modal space. The theoretical results are contrasted with two different numerical examples.

Computational Method of the Dynamic Response for Nonviscously Damped Structure Systems

AbstractNonviscous damping models in which the damping forces depend on the history of velocities via convolution integrals over some kernel functions have risen in many different subjects. The aim of this paper is to extend the Newmark method from viscously damped structure systems to nonviscously damped structure systems. The proposed analysis method for nonviscously damped structure systems is derived based on Newmark’s assumption of acceleration. The convolution integral term is calculated directly using the trapezoidal rule. The computational procedure of the proposed direct time integration method for nonviscously damped structure systems is given in detail. Finally, the dynamic responses of two nonviscously damped structure systems are computed using the proposed method. The accuracy and efficiency of the proposed method are discussed through comparison with another developed method.

Design sensitivity analysis of dynamic response of nonviscously damped systems

The sensitivity problem of dynamic analysis of linear nonviscously damped systems is considered. The assumed nonviscous damping forces depend on the past history of motion via convolution integrals over some kernel functions. The nonviscous damping model can be alternatively chosen from familiar viscoelastically damping structures and is considered as a further generalization of the familiar viscous damping. The computations of dynamic responses are reviewed for the purpose of design sensitivity analysis development. The dynamic response can be easily calculated using direct frequency response method and modal superposition method when the dynamic equation of motion of nonviscously damped systems is transformed into the frequency domain using the Laplace transform. It is shown that the dynamic response of nonviscously damped systems can be obtained using traditional modal analysis in a familiar manner used in undamped or viscously damped systems. The discrete Fourier transform and inverse discrete Fourier transform algorithms are also suggested to obtain the displacement in the time domain. Based on these expressions of dynamic response, the adjoint variable and direct differentiation methods, originally presented to obtain the dynamic response sensitivity of undamped or viscously damped systems, are both developed for efficiently and accurately calculating the sensitivity of dynamic response of nonviscously damped systems. Finally, some case studies are used to show the application, effectiveness and some characters of the derived formulas. The numerical sensitivity results show the sensitivity obtained using the developed methods are in excellent agreement with the finite difference results. However, the finite difference method suffers from computational inefficiency and possible errors.

Computation of Eigensolution Derivatives for Nonviscously Damped Systems Using the Algebraic Method

T HE computation of the eigensolution derivatives plays a significant role in dynamic model updating, structural design optimization, structural dynamic modification, damage detection andmany other applications. Themethods to calculate eigensolution derivatives are well established for undamped and viscous damped systems. These common methods can be divided into the modal method, Nelson’s method, and the algebraic method. Fox and Kapoor [1] first proposed the modal method for symmetric undamped systems by approximating the derivative of each eigenvector as a linear combination of all undamped eigenvectors. Later, Adhikari and Friswell [2] and Adhikari [3] extended the modal method to the more general asymmetric systems with viscous and nonviscous damping, respectively. To simplify the computation of eigensolution derivatives, Nelson [4] proposed a method, which requires only the eigenvector of interest by expressing the derivative of each eigenvector as a particular solution and a homogeneous solution for symmetric undamped systems. Later, Friswell and Adhikari [5] extended Nelson’s method to symmetric and asymmetric systemswith viscous damping. Recently, Adhikari and Friswell [6] extended Nelson’s method to symmetric and asymmetric nonviscously damped systems. However, Nelson’s method is lengthy and clumsy for programming. Lee et al. [7] derived an efficient algebraic method, which has a compact form to compute the eigensolution derivatives by solving a nonsingular linear system of algebraic equations for symmetric systems with viscous damping. Later, Guedria et al. [8] extended the algebraic method to general asymmetric systems with viscous damping. Recently, Chouchane et al. [9] wrote an excellent review of the algebraic method for symmetric and asymmetric systems with viscous damping and extended their method to the second-order and high-order derivatives of eigensolutions. In this note, the algebraic method will be extended to symmetric and asymmetric systems with nonviscous damping. The equations of motion describing free vibration of anN-degreeof-freedom (DOF) linear system with nonviscous (viscoelastic) damping can be expressed by [3,6]:

Dynamic Response Characteristics of a Nonviscously Damped Oscillator

The characteristics of the frequency response function of a nonviscously damped linear oscillator are considered in this paper. It is assumed that the nonviscous damping force depends on the past history of velocity via a convolution integral over an exponentially decaying kernel function. The classical dynamic response properties, known for viscously damped oscillators, have been generalized to such nonviscously damped oscillators. The following questions of fundamental interest have been addressed: (a) Under what conditions can the amplitude of the frequency response function reach a maximum value?, (b) At what frequency will it occur?, and (c) What will be the value of the maximum amplitude of the frequency response function? Introducing two nondimensional factors, namely, the viscous damping factor and the nonviscous damping factor, we have provided exact answers to these questions. Wherever possible, attempts have been made to relate the new results with equivalent classical results for a viscously damped oscillator. It is shown that the classical concepts based on viscously damped systems can be extended to a nonviscously damped system only under certain conditions.

Modal Analysis of Nonviscously Damped Beams

Linear dynamics of Euler–Bernoulli beams with nonviscous nonlocal damping is considered. It is assumed that the damping force at a given point in the beam depends on the past history of velocities at different points via convolution integrals over exponentially decaying kernel functions. Conventional viscous and viscoelastic damping models can be obtained as special cases of this general damping model. The equation of motion of the beam with such a general damping model results in a linear partial integro-differential equation. Exact closed-form equations of the natural frequencies and mode shapes of the beam are derived. Numerical examples are provided to illustrate the new results.

Calculation of Eigensolution Derivatives for Nonviscously Damped Systems

A method to calculate the derivatives of the eigenvalues and eigenvectors of multiple-degree-of-freedom damped linear dynamic systems with respect to arbitrary design parameters is presented. In contrast to the traditional viscous damping model, a more general nonviscous damping model is considered. The nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over given kernel functions. Because of the general nature of the damping, eigensolutions are generally complex valued, and eigenvectors do not satisfy the classical orthogonality relationship. The proposed method to calculate the eigenvector derivative depends only on the eigenvector concerned. Numerical examples are provided to illustrate the derived results. Nomenclature c j = constant associated with the derivative of u j D(s) = dynamic stiffness matrix d j = constant associated with the derivative of v j G(s) = Laplace transform of G(t) G(t) = nonviscous damping matrix K = stiffness matrix L[ ] = Laplace transform of [ ] M = mass matrix m = total number of eigenvectors N = number of degrees of freedom of the system n = number of relaxation parameters p = design parameter rank () = rank of a matrix s = Laplace domain parameter u j = jth eigenvector (t) = displacement vector v j = jth adjoint (left) eigenvector x j = vector associated with the derivative of u j yj = vector associated with the derivative of v j θ j = normalization constant for the jth eigenvector λ j = jth eigenvalue μk = relaxation parameters Subscripts () T = matrix transpose () � = derivative with respect to s

Qualitative dynamic characteristics of a non-viscously damped oscillator

This paper considers the linear dynamics of a single-degree-of-freedom non-viscously damped oscillator. It is assumed that the non-viscous damping force depends on the history of velocity via a convolution integral over an exponentially decaying kernel function. Classical qualitative dynamic properties known for viscously damped oscillators have been generalized to such non-viscously damped oscillators. The following questions of fundamental interest have been addressed: (i) under what conditions can a non-viscously damped oscillator sustain oscillatory motions? (ii) how does the natural frequency of a non-viscously damped oscillator compare with that of an equivalent undamped oscillator? and (iii) how does the decay rate compare with that of an equivalent viscously damped oscillator? Introducing two non-dimensional factors, namely, the viscous damping factor and the non-viscous damping factor, we provide answers to these questions. Wherever possible, attempts are made to relate the new results with equivalent classical results for a viscously damped oscillator.

Direct time-domain integration method for exponentially damped linear systems

Time-domain analysis of multiple-degree-of-freedom linear non-viscously damped systems is considered. It is assumed that the non-viscous damping forces depend on the past history of velocities via convolution integrals over exponentially decaying kernel functions. A direct time-domain integration method is proposed. The proposed approach is based on an extended state-space representation of the equations of motion. The state-space method, in turn, is based on introduction of a set of internal variables. The numerical method for the calculation of the displacements eliminates the need for explicit calculation of the velocities and usually large number of internal variables at each time step. This fact particularly makes this method numerically efficient. The proposed method is illustrated by two numerical examples.

Analysis of Asymmetric Nonviscously Damped Linear Dynamic Systems

Multiple-degree-of-freedom linear asymmetric nonviscously damped systems are considered. It is assumed that the nonviscous damping forces depend on the past history of velocities via convolution integrals over exponentially decaying kernel functions. An extended state-space approach involving a single asymmetric matrix is proposed. The nature of the eigensolutions in the extended state space has been explored. Some useful results relating the modal matrix in the extended state space and the modal matrix in the original space has been derived. Numerical examples are provided to illustrate the results.

Symmetric State-Space Method for a Class of Nonviscously Damped Systems

Multiple-degree-of-freedom linear nonviscously damped systems are considered. It is assumed that the nonviscousdampingforcesdependonthepasthistoryofvelocitiesviaconvolutionintegralsoverexponentiallydecaying kernel functions. The traditional state-space approach, well known for viscously damped systems, is extended to such nonviscously damped systems using a set of internal variables. Suitable numerical examples are provided to illustrate the proposed approach. I. Introduction V ISCOUSdampingisthemostcommonmodelforthemodeling of vibration damping in linear systems. This model, e rst introduced by Rayleigh, 1 assumes that the instantaneous generalized velocities are the only relevant variables that determine damping. Viscous damping models are used widely for their simplicity and mathematical convenience, even though the behavior of real structural materials is, at best, poorly mimicked by simple viscous models.Forthisreason,itiswellrecognizedthat,ingeneral,aphysically realistic model of damping will not be viscous. Damping models in whichthedissipativeforcesdependonanyquantityotherthantheinstantaneous generalized velocities are nonviscous damping models. Mathematically, any causal model that makes the energy dissipation functional nonnegative is a possible candidate for a nonviscous damping model. Clearly, a wide range of choice is possible, either based on the physics of the problem or by selecting a model a priori and e tting its parameters from experiments. Here, we will use a particular type of damping model that is not viscous, and throughout the paper the terminology nonviscous damping will refer to this specie c model only. Possiblythemostgeneralwaytomodeldampingwithinthelinear range is to use nonviscous damping models that depend on the past history of motion via convolution integrals over kernel functions. 2 The equations of motion of an N-degree-of-freedom linear system with such damping can be expressed by

Derivative of Eigensolutions of Nonviscously Damped Linear Systems

Derivatives of eigenvalues and eigenvectors of multiple-degree-of-freedom damped linear dynamic systems with respect to arbitrary design parameters are presented. In contrast to the traditional viscous damping model, a more general nonviscous damping model is considered. The nonviscous damping model is such that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions. Because of the general nature of the damping, eigensolutions are generally complex valued, and eigenvectors do not satisfy any orthogonality relationship. It is shown that under such general conditions the derivative of eigensolutions can be expressed in a way similar to that of undamped or viscously damped systems. Numerical examples are provided to illustrate the derived results.

Dynamics of Nonviscously Damped Linear Systems

This paper is aimed at extending classical modal analysis to treat lumped-parameter nonviscously damped linear dynamic systems. It is supposed that the damping forces depend on the past history of velocities via convolution integrals over some kernel functions. The traditional restriction of symmetry has not been imposed on the system matrices. The nature of the eigenvalues and eigenvectors is discussed under certain simplified but physically realistic assumptions concerning the system matrices and kernel functions. A numerical method for calculation of the right and left eigenvectors is suggested. The transfer function matrix of the system is derived in terms of the right and left eigenvectors of the second-order system. Exact closed-form expressions for the dynamic response due to general forces and initial conditions are presented. The proposed method uses neither the state-space approach nor additional dissipation coordinates. Suitable examples are given to illustrate the derived results.

Eigenrelations for Nonviscously Damped Systems

Multiple-degree-of-freedom linear dynamic systems with nonviscous damping is considered. The nonviscous damping is such that the damping forces depend on the past history of the velocities via convolution integrals over some kernel functions. For the sake of generality, the system matrices are allowed to be asymmetric. The mode-orthogonality relationships, known for undamped or viscously damped systems, have been generalized to such nonviscously damped systems. Some expressions are suggested for the normalization of the eigenvectors. A number of useful results relating the system matrices and the eigensolutions have been established. Numerical examples are provided to illustrate the derived results.

IDENTIFICATION OF DAMPING: PART 2, NON-VISCOUS DAMPING

In a companion paper (see pp. 43–61 of this issue), it was shown that when a system is non-viscously damped, an identified equivalent viscous damping model does not accurately represent the damping behaviour. This demands new methodologies to identify non-viscous damping models. This paper takes a first step, by outlining a procedure for identifying a damping model involving an exponentially decaying relaxation function. The method uses experimentally identified complex modes and complex natural frequencies, together with the knowledge of the mass matrix for the system. The proposed method and several related issues are discussed by considering numerical examples of a linear array of damped spring-mass oscillators. It is shown that good estimates can be obtained for the exponential time constant and the spatial distribution of the damping.